> >What logically could exist -- that is, what is not inherently self-> >contradictory -- has mathematical existence.>> The problem with this is that there could be two different> mathematical objects, A and B, such that neither is inherently> self-contradictory, but the existence of A contradicts the> existence of B. They can't, therefore, both exist.

Exactly so!

My favorite example being,(as someone just now pointed out half of),it is consistent with ZF that

1) there exists a set of cardinality strictly between N & R;

2) there exists a function on P(R) whose values are bijections between the argument and elements of N u {N,R}.